Does anyone know how to make an object(enemy) follow a path set by the player in a 2D side scroller?

I have just recently started coding and already ran into a problem. I want an enemy to chase the player, but by following what the player is doing. I want it to also increase the rate at which it follows the path that way it can actually catch the player. It doesn’t matter whether the code is in C# or Javascript. Thank you for your time and help.

var nextUsage;
var delay = 0;
var target : Transform;
private var thisTransform : Transform;
private var velocity : Vector2;

function Start()
{
nextUsage = Time.time + delay;
thisTransform = transform;
}

function Update()
{
if (Time.time > nextUsage)
{
nextUsage = Time.time + delay;
thisTransform.position.x = (target.position.x - 60/Time.time);
thisTransform.position.y = (target.position.y);
}
}

Thank you very much. I really appreciate all the help. I will look into more AI steering behaviors to advance the enemy.